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Robust decentralized control in power systems Claudio De Persis Institute of Engineering and Technology J.C. Willems Center for Systems and Control PowerWeb Lunch Lecture TU Delft, December 13, 2018 Joint work with Weitenberg (RUG), Jiang-Mallada (Johns Hopkins), Zhao (NREL), D¨ orfler (ETH)

Robust decentralized control in power systems · Robust decentralized control in power systems Claudio De Persis Institute of Engineering and Technology J.C. Willems Center for Systems

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Page 1: Robust decentralized control in power systems · Robust decentralized control in power systems Claudio De Persis Institute of Engineering and Technology J.C. Willems Center for Systems

Robust decentralized control in power systems

Claudio De Persis

Institute of Engineering and TechnologyJ.C. Willems Center for Systems and Control

PowerWeb Lunch LectureTU Delft, December 13, 2018

Joint work with Weitenberg (RUG), Jiang-Mallada (JohnsHopkins), Zhao (NREL), Dorfler (ETH)

Page 2: Robust decentralized control in power systems · Robust decentralized control in power systems Claudio De Persis Institute of Engineering and Technology J.C. Willems Center for Systems

AC power systems

• Power system = networkof generation, loads,transmission lines

• Power system control =maintain system securityat minimal cost

• Basic security requirement= keeping frequencyaround nominal value

1 / 22

Page 3: Robust decentralized control in power systems · Robust decentralized control in power systems Claudio De Persis Institute of Engineering and Technology J.C. Willems Center for Systems

Frequency controlFrequency control: Basics

• Any instantaneous load-generation imbalance results in afrequency deviation from the nominal one (50-60 Hz)

• Small load changes on a fast time scale are dealt with theAutomatic Generation Control (AGC)

2 / 22

Page 4: Robust decentralized control in power systems · Robust decentralized control in power systems Claudio De Persis Institute of Engineering and Technology J.C. Willems Center for Systems

The three control layers

Fig.�6.� Inertial,�primary�frequency�controls,�and�AGC�(secondary)�response�(figure�courtesy�Pouyan�Pourbeik�of�EPRI).�

by� individual� power� plants� adjusting� their� power� levels� in�response� to� requests� from�the�systems�operator.�

B.� Motivation� for�Active�Power�Control� in�Wind�Turbines�

Active� power� control� (APC)� is� the� purposeful� control� of�the�real�power�output�of�a�wind�turbine�or�collection�of�wind�turbines�in�order�to�assist�in�balancing�total�power�generated�on�the�grid�with�total�power�consumed.�Real�(also�known�as�“active”)� power� is� the� portion� of� energy� flow� which� results�in� a� net� transfer� of� energy;� this� is� in� contrast� to� imaginary�or� reactive� power� flow,� which� reverses� direction� over� each�cycle� with� no� net� energy� transfer.� While� reactive� power�may� be� controlled� through� the� turbine’s� power� electronics�(even�when�the�turbine�is�offline),�real�power�is�regulated�by�changing�the�actual�amount�of�power�delivered�to�the�utility�grid.� In� doing� so,� APC� may� allow� wind� power� to� provide�important� ancillary� services� to� the� electrical� grid� and� assist�in�maintaining�an�acceptable� frequency.�

Intrinsically,� wind� turbines� do� not� provide� APC� services�and�they�have�not�historically�been�required�to�provide�such�responses� [21].� Most� modern� turbine� generators� are� decou-pled� from�grid� frequency� through�power� electronics� (type�3�or� 4),� as� described� in� Section� II-B.� Therefore,� the� inertia�of� the� generator� and� the� turbine� rotor� do� not� automatically�participate� in� the� grid� inertial� response� as� would� traditional�synchronous�generators.�Further,�because�of�this�decoupling,�changes� in� grid� frequency�do�not� elicit� an� automatic� gover-nor� response� that� is� common� with� conventional� generation�sources.�

Increasingly,�there�are�two�main�motivations�for�why�APC�should� be� provided� by� wind� turbines.� The� first� motivation�is� that� regulation� is�essential� for�maintaining�grid� frequency�and� as� wind� penetration� increases� it� can� provide� key� sup-port� in� maintaining� the� required� balance.� A� FERC/LBNL�(Federal�Energy�Regulatory�Commission/Lawrence�Berkeley�National�Laboratory)�study�discusses�a�recent�decline�in�grid�frequency�response,�and�states�that�although�increasing�wind�penetration� is� not� the� cause,� frequency� response� could� be�improved� through� the� expanded� use� of� frequency� control�capabilities�in�wind�turbines�[22,�23].�Further,�a�recent�study�by� the� IPCC� (Intergovernmental� Panel� on� Climate� Change)�

found�that�although�low�to�medium�wind�penetrations�(up�to�20%�of�annual�demand)�poses�no�“insurmountable�technical�barriers,”� higher� levels� could� require� additional� flexibility�options� such� as� greater� use� of� wind� power� curtailment� and�output�control�[24].�As�wind�energy�penetration�increases,�it�is� potentially� displacing� regulation-providing� generators.� At�very� high� wind� penetration� levels� it� becomes� necessary� for�wind�turbines�to�also�provide�these�services�[21]�as�it�has�for�several�European�grid�operators.�

In�countries�and�regions�with�relatively� isolated�grids�and�relatively� large� levels� of� wind� penetration,� participation� in�grid�frequency�regulation�by�wind�turbines�and�wind�plants�is�crucial.�A�2010�report�issued�by�Project�UpWind�[2]�indicates�that� replacing� conventional� generation� sources� with� a� large�percentage�of�wind�power�not�capable�of�active�power�control�can� potentially� have� significant� impacts� on� stability� of� grid�frequency.�This�effect�is�pronounced�on�island�grids�with�low�levels�of�interconnectivity�to�other�grids,�such�as�many�of�the�Greek� isles� [25].�

The�necessity�of�participation�in�grid�frequency�regulation�by�wind�turbines�is�reflected�in�the�requirements�and�regula-tions�put�on�wind�plants�by�TSOs�in�regions�with�high�levels�of�wind�penetration�or�relatively�isolated�grids.�For�example,�the� Irish� grid� code� requires� that� wind� plants� have� active�power� curtailment� capabilities,� and� outlines� specific� active�power�generation�set-points�as�a�function�of�available�power�in�the�event�of�a�frequency�deviation�[26].�This�code�further�specifies� a� minimum� response� rate� for� individual� turbines�of� 1� %� of� rated� power� per� second.� Elsewhere,� Denmark’s�TSOs� Eltra� and� Elkraft� require� that� wind� plants� be� able�to� track� a� reserve� power� offset� and� track� reference� power�levels� generated� by� the� system� operator� [27].� In� Canada,�Hydro-Quebec�requires�that�wind�plants�rated�above�10�MW�have� the� ability� to� modify� their� active� power� output� for� at�least� 10� seconds� in� response� to� grid� frequency� deviations�greater�than�0.5�Hz� [28].�Additionally,�the�Spanish�TSO,�Red�Electrica,� mandates� that� wind� plants� respond� to� frequency�deviations� with� proportional� control� of� active� power� output�within�a� range�of�percentages�of� rated�power�[29].�

The� second� motivation� for� active� power� control� by� wind�turbines� is� the�potential� to� increase� the�profitability�of�wind�plants�by�enabling�participation�in�ancillary�service�markets.�A� recent� study� by� Kirby� demonstrates� a� potential� for� wind�plants�to�“increase�their�own�profits�by�providing�regulation”�[30].�

C.� Industry�Activity� in�Support�of�APC�

The� various� active� power� control� requirements� laid� out�by�grid�operators�have�motivated�technological�development�by� wind� turbine� manufacturers.� These� developments� have�been�made�at�both� individual� turbine�and�wind�plant� scales.�Technologies� to� provide� response� in� all� of� the� inertial,� pri-mary,�and�secondary�time�scales�have�been�developed�at� the�individual� turbine� level.�For� example,�Siemens�has�patented�a�method� for�dynamically�modifying�an� individual� turbine’s�active�power�output�in�response�to�grid�frequency�deviations�

6

[figure EPRI]

• Primary control counteracts initial frequency drop and implementedvia local droop control of turbine governors

• Secondary AGC is centralized and uses integral control to restorefrequency

3 / 22

Page 5: Robust decentralized control in power systems · Robust decentralized control in power systems Claudio De Persis Institute of Engineering and Technology J.C. Willems Center for Systems

Conventional operational strategy

Central control authority

• AGC is implemented with a central regulator

• Frequency deviation ω is measured at low-voltage network andintegrated to generate the regulator output signal p

T p = ω

• The incremental contribution of the individual generatingunits to the total generation is obtained via participationsfactors K−1

i

ui = −K−1i p, i = 1, 2, . . . , n

4 / 22

Page 6: Robust decentralized control in power systems · Robust decentralized control in power systems Claudio De Persis Institute of Engineering and Technology J.C. Willems Center for Systems

Conventional operational strategy

• The conventional strategy is developed for conventionalgenerators which have high inertia, hence abrupt changes arebetter absorbed by the system, thus easing the task offrequency restoration

• Renewable generation leads to significant reduction of inertia,hence to a more volatile network, which challenges existingcontrol schemes

5 / 22

Page 7: Robust decentralized control in power systems · Robust decentralized control in power systems Claudio De Persis Institute of Engineering and Technology J.C. Willems Center for Systems

Distributed control

An answer to this challenge has leveraged the use of localcontrollers that cooperate over a communication network

ICT for Network Systems

PHYS

ICAL

NETW

ORK

COMMUN

ICATION

NETW

ORK

!1 = "1M1"1 = !A1"1 + u1 ! Pl1 + Pe1

y1 = "1

...

!n = "nMn"n = !An"n + un ! Pln + Pen

yn = "n

!1 =!

j!N1(!j ! !1) ! q"1

1 "1u1 = q"1

1 "1

...

!n =!

j!Nn(!j ! !n) ! q"1

n "nun = q"1

n "n

Pe1

Pen

!

!

"1

u1

"n

un

15 / 32

• Semi-centralized

Tp =∑

i ωi , ui = −K−1i p

• Distributed averaging integral

Ti pi =∑

j∈N ci

(pj − pi ) + K−1i ωi

ui = −K−1i pi

Yet, due to security, robustness and economic concerns, it isdesirable to regulate the frequency without relying oncommunication

6 / 22

Page 8: Robust decentralized control in power systems · Robust decentralized control in power systems Claudio De Persis Institute of Engineering and Technology J.C. Willems Center for Systems

Power network

Lossless, network-reduced power system with n generating units

POWER NETWORK

gener. unit 6 gener. unit 7 gener. unit 8

gener. unit 1 gener. unit 2

gener. unit 5gener. unit 4

gener. unit 3

generating unit ifrequency deviation ωi

controll. power inj. uiuncontroll. power inj. P?i

generating unit jfrequency deviation ωj

controll. power inj. ujuncontroll. power inj. P?j

power

transmission

[figure Stegink]

7 / 22

Page 9: Robust decentralized control in power systems · Robust decentralized control in power systems Claudio De Persis Institute of Engineering and Technology J.C. Willems Center for Systems

Frequency dynamics

θi = ωi , Mi ωi = −Aiωi −∑

j∈Ni

γij︷ ︸︸ ︷ViVjBij sin(θi − θj) + ui − P?i

Local measurements: ωi

• Swing equations

• ωi frequency deviation

• θi phase angle deviation

• voltages Vi constant

• purely inductive linesBij = Bji

• mechanical equivalent ⇒Energy function:H = −1

2

∑i 6=j

BijViVj cos(θi − θj) +1

2

∑i

Miω2i

M1 u1

ω1

M2

u2

ω2

M3

u3ω3

M4

u4

ω4

M5

u5

ω5

M6

u6 ω6M7

u7

ω7

M8

u8

ω8

γ12

γ23

γ34

γ45

γ56

γ67

γ81

γ36

[figure Stegink]8 / 22

Page 10: Robust decentralized control in power systems · Robust decentralized control in power systems Claudio De Persis Institute of Engineering and Technology J.C. Willems Center for Systems

Synchronization frequency control

θi = ωi

Mi ωi = −Diωi + ui + P?i −∑

j BijEiEj sin(θi − θj)

• Synchronous solution ωi = ωsync

• Synchronous frequency ωsync =∑

i P?i +

∑i ui∑

i Di

Case n = 2

M1ω1 = −D1ω1 + u1 + P?1 − B12E1E2 sin(θ1 − θ2)

M2ω2 = −D2ω2 + u2 + P?2 − B21E2E1 sin(θ2 − θ1)

If ω1 = ω2 = ωsync = const, summing up

0 = −(D1 + D2)ωsync + u1 + u2 + P?1 + P?

2

• Zero frequency deviation 0 =∑

i P?i +

∑i ui

9 / 22

Page 11: Robust decentralized control in power systems · Robust decentralized control in power systems Claudio De Persis Institute of Engineering and Technology J.C. Willems Center for Systems

Optimal frequency restoration

Manifold choices of u?i to achieve

0 =∑

i

P?i +∑

i

u?i

Optimal dispatch problem

minimizeu∈Rn

∑i aiu

2i

subject to∑

i P∗i +

∑i ui = 0

Solution

u?i = −a−1i

∑i P

?i∑

i ai

Fair proportional sharing

aiu?i = aju

?j ∀i , j

Optimal frequency restorationGiven unknown P?i , design

ui (ωi )

that stabilizes the powersystem model to

(θ?i , ω?i = 0, u?i )

10 / 22

Page 12: Robust decentralized control in power systems · Robust decentralized control in power systems Claudio De Persis Institute of Engineering and Technology J.C. Willems Center for Systems

Fully decentralized frequency control

Ti pi = ωi

ui = −pi

• No communicationrequired

• Frequency regulationω → 0

IEEE 39-node ‘New England’ benchmark network

0 20 40 60 80

Time (s)

59.6

59.7

59.8

59.9

60

Fre

quen

cy (

Hz)

no noise

noisy

• Frequency at G1

• Noisy measurementsωi + ηi

• Non-zero mean noise η

• Noise bound η = 0.01Hz

11 / 22

Page 13: Robust decentralized control in power systems · Robust decentralized control in power systems Claudio De Persis Institute of Engineering and Technology J.C. Willems Center for Systems

Fully decentralized frequency control

• No optimalityu → u(p(0), θ(0)) 6= u?

Active power output of all generators (no noise)

0 20 40 60 80

Time (s)

-50

0

50

100

150

Pow

er

(MW

)

0 20 40 60 80

Time (s)

-50

0

50

100

150

Pow

er

(MW

)

Active power output of all generators (noise)

• Unstable behavior

• Steady state

0 ≈ Ti pi = ωi + ηi 6=0 ≈ Tj pj = ωj + ηj

12 / 22

Page 14: Robust decentralized control in power systems · Robust decentralized control in power systems Claudio De Persis Institute of Engineering and Technology J.C. Willems Center for Systems

Leaky integral control

Ti pi = ωi−Kipiui = −pi

• No communicationrequired

• Synchronousfrequency

ωsync =

∑i P

?i∑

i Di +∑

i K−1i

• Banded frequencyregulation

∑i

K−1

i ≥∑

i P?i

ε−∑i

Di ⇒ |ωsync| ≤ ε

0 20 40 60 80

Time (s)

59.6

59.7

59.8

59.9

60

Fre

quen

cy (

Hz)

no noise

noisy

Leaky integral control Ti = 0.05s, Ki = 0.005 for G3, G5, G6, G9, G10, Ki = 0.01

for the others

0 20 40 60 80

Time (s)

59.6

59.7

59.8

59.9

60

Fre

quen

cy (

Hz)

no noise

noisy

Decentralized pure integral control13 / 22

Page 15: Robust decentralized control in power systems · Robust decentralized control in power systems Claudio De Persis Institute of Engineering and Technology J.C. Willems Center for Systems

Leaky integral control

Ti pi = ωi−Kipiui = −pi

• Steady-state

u?i = −K−1i ωsync

• Power sharing

Kiu∗i = Kju

∗j

• Approx steady-state optimality

minimizeu∈Rn∑

i Kiu2i

subject to∑

i P∗i +

∑i (1 + DiKi )ui = 0

0 20 40 60 80

Time (s)

-50

0

50

100

150

Pow

er

(MW

)

Leaky integral control Ti = 0.05s, Ki = 0.005 for G3, G5, G6,

G9, G10, Ki = 0.01 for the others

0 20 40 60 80

Time (s)

-50

0

50

100

150

Pow

er

(MW

)

Decentralized pure integral control

14 / 22

Page 16: Robust decentralized control in power systems · Robust decentralized control in power systems Claudio De Persis Institute of Engineering and Technology J.C. Willems Center for Systems

Leaky integral control

Ti pi = ωi + ηi−Kipiui = −pi

• Noisy measurements ωi + ηi

• Non-zero mean noise η

• Noise bound η = 0.01Hz

0 20 40 60 80

Time (s)

-50

0

50

100

150

Pow

er

(MW

)

Leaky integral control Ti = 0.05s, Ki = 0.005 for G3, G5, G6,

G9, G10, Ki = 0.01 for the others

Robust frequency regulation (ISS)

‖x(t)‖2 ≤ λe−αt‖x(0)‖2 + γ( supt∈R≥0

‖η(t)‖)2

where λ, α, γ are positive constants and

x = col(δ − δ?, ω − ω?, p − p?)

measures the deviation from the synchronous solution15 / 22

Page 17: Robust decentralized control in power systems · Robust decentralized control in power systems Claudio De Persis Institute of Engineering and Technology J.C. Willems Center for Systems

Impact of control parameters

Tuning of the gains Ki

Ki = k for G3 G5 G6 G9 G10Ki = 2k for othersTi = τ = 0.05s

As k ↗• Noise-free steady-state frequency

error ↗• α↗ implies convergence time ↘• γ ↘ implies RMSE ↘

0 0.002 0.004 0.006 0.008 0.010

0.1

0.2

Fre

qu

ency

err

or

(Hz)

0 0.002 0.004 0.006 0.008 0.0110

20

30

40

50

Co

nv

erg

ence

tim

e (s

)

0 0.002 0.004 0.006 0.008 0.01

Inverse DC gain k

3

4

5

6

7

Fre

qu

ency

RM

SE

(H

z)

×10-3

Increasing gains Ki leads to

• reduced accuracy in frequency regulation• faster response• increased robustness to noise

16 / 22

Page 18: Robust decentralized control in power systems · Robust decentralized control in power systems Claudio De Persis Institute of Engineering and Technology J.C. Willems Center for Systems

Impact of control parameters

Tuning of the time constants Ti

Ki = 0.005 for G3 G5 G6 G9 G10Ki = 0.01 for the othersTi = τ

As τ ↗• α↘ implies convergence time ↗• γ ↘ implies RMSE ↘

0 0.02 0.04 0.06 0.08 0.1

10

20

30

Conver

gen

ce t

ime

(s)

0 0.02 0.04 0.06 0.08 0.1

Time constant τ (s)

4

6

8

10

Fre

quen

cy R

MS

E (

Hz)

×10-3

Increasing time constants Ti leads to

• slower response

• increased robustness to noise

17 / 22

Page 19: Robust decentralized control in power systems · Robust decentralized control in power systems Claudio De Persis Institute of Engineering and Technology J.C. Willems Center for Systems

Tuning recommendations

0 0.002 0.004 0.006 0.008 0.010

0.1

0.2

Fre

qu

ency

err

or

(Hz)

0 0.002 0.004 0.006 0.008 0.0110

20

30

40

50

Co

nv

erg

ence

tim

e (s

)

0 0.002 0.004 0.006 0.008 0.01

Inverse DC gain k

3

4

5

6

7

Fre

qu

ency

RM

SE

(H

z)

×10-3

0 0.02 0.04 0.06 0.08 0.1

10

20

30

Co

nv

erg

ence

tim

e (s

)

0 0.02 0.04 0.06 0.08 0.1

Time constant τ (s)

4

6

8

10

Fre

qu

ency

RM

SE

(H

z)

×10-3

• Fix ratios between K−1i from

generating units operationcosts

u?i = −K−1i ωsync

• Fix∑

i K−1i for banded

frequency regulation

i

K−1i ≥

∑i P

?i

ε−∑

i

Di

• Fix Ti to strike a trade-offbetween frequency rejectionrate and noise rejection

Ti ↗ ⇒ α↘ and γ ↘18 / 22

Page 20: Robust decentralized control in power systems · Robust decentralized control in power systems Claudio De Persis Institute of Engineering and Technology J.C. Willems Center for Systems

A further comparison

Decentralized inte-

gralDecentralized leaky DAI

Frequency

0 20 40 60 80

Time (s)

59.6

59.7

59.8

59.9

60

Fre

quen

cy (

Hz)

no noise

noisy

0 20 40 60 80

Time (s)

59.6

59.7

59.8

59.9

60

Fre

quen

cy (

Hz)

no noise

noisy

0 20 40 60 80

Time (s)

59.6

59.7

59.8

59.9

60

Fre

quen

cy (

Hz)

no noise

noisy

Power

no noise

0 20 40 60 80

Time (s)

-50

0

50

100

150

Pow

er

(MW

)

0 20 40 60 80

Time (s)

-50

0

50

100

150

Pow

er

(MW

)

0 20 40 60 80

Time (s)

-50

0

50

100

150

Pow

er

(MW

)

Power

with noise

0 20 40 60 80

Time (s)

-50

0

50

100

150

Pow

er

(MW

)

0 20 40 60 80

Time (s)

-50

0

50

100

150

Pow

er

(MW

)

0 20 40 60 80

Time (s)

-50

0

50

100

150

Pow

er

(MW

)

DAI Ti pi =∑

j∈N ci

(pj − pi ) + K−1i ωi ui = −K−1

i pi 19 / 22

Page 21: Robust decentralized control in power systems · Robust decentralized control in power systems Claudio De Persis Institute of Engineering and Technology J.C. Willems Center for Systems

Robust stability

Proof is Lyapunov-based, using a strict Lyapunov function

W = U(δ)− U(δ∗)−∇U(δ∗)>(δ − δ∗)

+1

2(ω − ω∗)>M(ω − ω∗) +

1

2(p − p∗)>T (p − p∗)

+ε(∇U(δ)−∇U(δ∗))>M(ω − ω∗).

• U(δ) = −12

∑i 6=j BijViVj cos(δi − δj) potential energy

• W with ε = 0 is the “shifted” energy function H + 12p>Tp

• For sufficiently small ε, W is strictly decreasing along thesolutions

• This allows for quantification of robustness to noise

20 / 22

Page 22: Robust decentralized control in power systems · Robust decentralized control in power systems Claudio De Persis Institute of Engineering and Technology J.C. Willems Center for Systems

Conclusions

A fully decentralized stabilizing integral control for achieving robustnoise-rejection, satisfactory steady-state regulation, desirable transientperformance

• These objectives are not aligned and trade-offs must be found

• Tuning guidelines are provided

• Resulting time constants Ti/Ki compatible with actuator response time

• Low-pass filter compares favourably wrt droop (noise rejection)

Future work

• Lead compensators could improve transient performance

• Extension more accurate physical models

• Impact of topology on the diffusion of noise and scalability

Reference

Weitenberg, Jiang, Zhao, Mallada, De Persis, Dorfler (2018). Robust decentralized secondary frequency control inpower systems: merits and trade-offs. IEEE Transactions on Automatic Control, in press, available asarXiv:1711.07332•Weitenberg, De Persis, Monshizadeh (2018). Exponential convergence under distributed averaging integralfrequency control. Automatica, 98, 103-113.?Weitenberg, De Persis (2018). Robustness to noise of distributed averaging integral controllers. Systems &Control Letters, 119, 1-7.

21 / 22

Page 23: Robust decentralized control in power systems · Robust decentralized control in power systems Claudio De Persis Institute of Engineering and Technology J.C. Willems Center for Systems

Thank you!

22 / 22